Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).
Generalities on Completions
Suppose we have a filtered abelian group with a descending filtration of subgroups . Because of this, we can consider “Cauchy sequences” and “convergence:”
Definition 1The sequence , is Cauchy if for each , there exists large enough that
The sequence converges to if for each , there exists large enough that
If we endow with with the pseudo-metric
then Cauchy sequences and convergence in this language reduce to the corresponding notions in the classical language.
As one constructs the reals from the rationals, one can take a completion:
Definition 2The completion of is defined as the abelian group consisting of equivalence classes of Cauchy sequences , where
This is a group under pointwise addition.
is said to be complete if ; this is equivalent to the assertion that each Cauchy sequence has a unique limit in . Note that the existence of a limit is not enough; the canonical map sending to the equivalence class of the sequence is injective if and only if is Hausdorff.
I like formulating the definition in terms of Cauchy sequences, but there is also a more algebraic construction in terms of inverse limits. Given a sequence of groups , with homomorphisms for such that for , we define the inverse limit to consist of the subset of consisting of strings such that . This is actually a group. If the are rings and the ring-homomorphisms, is a ring too.
The inverse limit construction is perhaps best thought of in more generality, as an example of a categorical limit. Then inverse limits can be defined in arbitrary categories, although they need not exist.
Proposition 3 We have , where the inverse limit is taken to the canonical maps for .
This can be checked directly. Fix . Given a Cauchy sequence , the images eventually stabilize to some , so one gets an element . For the reverse, given an element , pick liftings of to to get a Cauchy sequence. Since we are working modulo equivalence classes, it follows that we have an isomorphism.
Some books seem to define the completion just using the inverse limit and do everything more or less algebraically, but I think it is also useful to think topologically, which can make proofs easier; this tends to be emphasized in material on the -adic numbers.
All the same, it is of course also useful to think algebraically, and the next result is perhaps best thought of using the definition . The next result will be useful in proving that completion preserves Noetherianness for rings:
Proposition 4 There is a filtration with . In the resulting topology, is complete.
Define to consist of elements of the inverse limit with for ; this is implied by . The map is just projection on the th coordinate. It is easily seen to be both surjective and injective.
Since , we find that , so is complete.
Note that the result implies moreover .
I didn’t get as far as I would have hoped in this post, but I will keep going next week on completions of rings.